The all-electron GW method based on WIEN2k: Implementation and applications.

The all-electron GW method based on WIEN2k:
Implementation and applications.
Ricardo I. G´omez-Abal
Fritz-Haber-Institut of the Max-Planck-Society
Faradayweg 4-6, D-14195, Berlin, Germany
15th. WIEN2k-Workshop
March, 29th. 2008
R. G´omez-Abal (FHI-Berlin) GW@Wien2k Vienna 2008 1 / 54

Outline
Outline
1 Introduction
2 Implementation
GW@wien2k
Convergence Tests
3 Results
Bandgaps
Bandstructures
Macroscopic Dielectric Constant
4 Core-valence interaction
Bandgaps
Semicore States
5 f-electron systems
6 Conclusions

Introduction
Density Functional Theory
E ⇐⇒ ρ
Kohn-Sham scheme
interacting electrons ﬁctious particles
non interacting
Condition:
n(r) =
occ
i
|Ψi (r)|2

Introduction
Density Functional Theory
Kohn-Sham equation
[T + Vext + VH + Vxc]Ψi = ǫi Ψi
Vxc
local
energy independent
hermitian
ǫi :
Lagrange multipliers
No physical meaning
Exception: Highest occupied ǫi = −I
Fast.
Good structural properties.
Excitation spectra??
E ⇐⇒ ρR. G´omez-Abal (FHI-Berlin) GW@Wien2k Vienna 2008 4 / 54

Introduction
The Bandgap Problem
LDA vs. experimental bandgaps
Si
GaAs
zGaN
ZnS
C
CaO
MgO
NaCl
0 2 4 6 8
Eg
exp
[eV]
0
2
4
6
8Eg
LDA
[eV]
Up to 50% underestimation

Introduction
Many-Body Theory
Quasiparticle equation
[T + Vext + VH]Ψi (r) + Σ(r, r′
; ǫi )Ψi (r′
)d3
r′
= ǫi Ψi (r)
Σ
non local
energy dependent
non hermitian
ǫi :
Poles of the Green’s Function
ǫi =
E(N) − E(N − 1, i) ǫi < EF
E(N + 1, i) − E(N) ǫi > EF
Formally correspond to the excitation spectra

Introduction
Many Body Theory (cont.)
The Self-Energy (Σ)
Hedin, 1965: Expansion in terms of the dynamically screened Coulomb
potential (W ).
Fast convergence.
First order:
Σ(r, t, r′
, t′
) = G(r, t, r′
, t′
)W (r, t, r′
, t′
)
Simplest approximation including dynamical correlation eﬀects

Introduction
Many Body Theory (cont.)
The Screened Coulomb potential (W )
W (r1, r2; ω) = ε−1
(r1, r3; ω)v(r3, r2)dr3
ε(r1, r2; ω) =1 − v(r1, r3)P(r3, r2; ω)dr3
P(r1, r2; ω) = −
i
2π
G(r1, r2; ω + ω′
)G(r2, r1; ω‘)dω′
Requires selfconsistency with the QP equation!

Introduction
Perturbative treatment
G0W0
[T + Vext + VH]Ψi (r)+
Σ(r, r′
; ǫi ) Ψi (r′
)d3
r′
= ǫi Ψi (r)

Introduction
G0W0
[T + Vext + VH + Vxc]Ψi (r)+
Σ(r, r′
; ǫi ) − Vxc(r′
)δ(r − r′
) Ψi (r′
)d3
r′
= ǫi Ψi (r)

Introduction
G0W0
Σ(r, r′
)δ(r − r′
) Ψi (r′
)d3
r′
= ǫi Ψi (r)
First order correction to ǫKS
nk :
ǫqp
nk = ǫKS
nk + ∆ǫnk
∆ǫnk = ℜ( Ψnk(r)|Σ(r, r′
, ǫqp
nk)|Ψnk(r′
) ) − Ψnk(r)|Vxc |Ψnk(r)

Introduction
G0W0
Σ(r, r′
)δ(r − r′
) Ψi (r′
)d3
r′
= ǫi Ψi (r)
First order correction to ǫKS
nk :
ǫqp
nk = ǫKS
nk + ∆ǫnk
∆ǫnk = ℜ( Ψnk(r)|Σ(r, r′
, ǫqp
nk)|Ψnk(r′
) ) − Ψnk(r)|Vxc |Ψnk(r)
Σ calculated in the GW approximation.

Introduction
Introduction
G0W0 equations
G0(r1, r2; ω) =
occ
nk
Ψnk(r1)Ψ∗
nk(r2)
ω − ǫnk − iη
+
unocc
nk
Ψnk(r1)Ψ∗
nk(r2)
ω − ǫnk + iη
P(r1, r2; ω) = −
i
2π
G0(r1, r2; ω + ω′
)G0(r2, r1; ω‘)dω′
ε(r1, r2; ω) =1 − v(r1, r3)P(r3, r2; ω)dr3
W0(r1, r2; ω) = ε−1
(r1, r3; ω)v(r3, r2)dr3
Σ(r1, r2; ω) =
i
2π
G0(r1, r2; ω + ω′
)W0(r2, r1; ω‘)dω′

Implementation GW@wien2k
Implementation

Implementation Basis Functions
The Polarization
P(r1, r2, ω) =
X
n,m,k,k′
Ψnk(r1)Ψ∗
mk′ (r1)Ψ∗
nk(r2)Ψmk′ (r2)F(ǫnk, ǫmk′ ; ω)

The basis functions
P(r1, r2, ω) =
X
n,m,k,k′
Ψnk(r1)Ψ∗
mk′ (r1)Ψ∗
nk(r2)Ψmk′ (r2)F(ǫnk, ǫmk′ ; ω)
Deﬁnition
χq
i (r) =



Ra
eik·(R+ra)
υNL(r)YLM (ˆr) r ∈ MT-spheres
1
√
V G
Si,Gei(q+G)·r
r ∈ Interstitial
1 F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 237 (1998).
2 T. Kotani and M. van Schilfgaarde, Sol. State. Comm. 121, 461 (2002).
MT
Sphere
Interstitial

The basis functions
Obtaining the radial functions
For each L take ul (r)ul′ (r) such that |l − l′| ≤ L ≤ l + l′
Calculate the overlap matrix:
Oa
ll′,l1l′
1
=
RMT
0
ul (r)ul′ (r)ul1
(r)ul′
1
(r)r2
dr
Solve the secular equation:
Oa
ll′,l1l′
1
− λnδll1
δl′l′
1
cn,l1l′
1
= 0
If λn ≥ λtol then:
υnL(r) =
ll′
cn,ll′ ul (r)ul′ (r)

The basis functions
The matrix elements
Mi
nm(k, q) ≡
Ω
˜χq
i (r)Ψm,k−q(r)
∗
Ψnk(r)d3
r
Tests
Visual
Ψn,k(r)Ψ∗
m,k−q(r) =
i
Mi
nm(k, q)˜χq
i (r)
Completeness
∆ =
R
|Ψn,k(r)Ψ∗
m,k−q(r)−
P
i Mi
nm(k,q)˜χq
i (r)|2d3r
R
|Ψn,k(r)Ψ∗
m,k−q(r)|2d3r
∆ = 1 − i |Mi
nm(k, q)|2
|Ψn,k(r)Ψ∗
m,k−q(r)|2d3r

Basis Functions: Tests
k = Γ, k′
= X, n = 4 m = 5
At1
RMT1
RMT2
At2
r
-0.04
-0.02
0
0.02
Ψnk
Ψmk’
Exact
Fit
Ψn,k(r)Ψ∗
m,k−q(r) =
X
i
Mi
nm(k, q)˜χq
i (r)

k = Γ, k′
= π
a (1, 0, 0), n = 2 m = 6
At1
RMT1
RMT2
At2
r
-0.01
0
0.01
Ψnk
Ψmk’
Exact
Fit
Ψn,k(r)Ψ∗
m,k−q(r) =
X
i
Mi
nm(k, q)˜χq
i (r)

k = 3π
2a (1, 1, −1), k′
= π
2a (1, −1, 1), n = 2, m =
6
At1
RMT1
RMT2
At2
r
-0.1
-0.05
0
0.05
0.1
Ψnk
Ψmk’
Exact
Fit
Ψn,k(r)Ψ∗
m,k−q(r) =
X
i
Mi
nm(k, q)˜χq
i (r)

k = Γ, k′
= π
a (1, 0, 0), n = 1, m = 5
At1
RMT1
RMT2
At2
r
0
0.5
1Ψnk
Ψmk’
Exact
Fit
Ψn,k(r)Ψ∗
m,k−q(r) =
X
i
Mi
nm(k, q)˜χq
i (r)

Basis Functions: Completeness test.
MT-Spheres
20 40 60 80 100 120 140 160 180
Number of spherical basis functions per atom
1e-06
1e-05
1e-04
0.001
0.01
0.1
Relativeerror
tol.=1e-2
tol.=1e-3
tol.=1e-4
tol.=1e-5
tol.=1e-7
tol.=1e-2
tol.=1e-3
tol.=1e-4
tol.=1e-5
lmax=2 lmax=3lmax=1
tol.=1e-2
tol.=1e-3
tol.=1e-5
Minimum
Maximum

MT-Spheres
20 40 60 80 100 120 140 160 180
1e-06
1e-05
1e-04
0.001
0.01
0.1
Relativeerror
tol.=1e-2
tol.=1e-3
tol.=1e-4
tol.=1e-5
tol.=1e-7
tol.=1e-2
tol.=1e-3
tol.=1e-4
tol.=1e-5
lmax=2 lmax=3lmax=1
tol.=1e-2
tol.=1e-3
tol.=1e-5
Minimum
Maximum
Remarks
Decreases with lmax .

MT-Spheres
20 40 60 80 100 120 140 160 180
1e-06
1e-05
1e-04
0.001
0.01
0.1
Relativeerror
tol.=1e-2
tol.=1e-3
tol.=1e-4
tol.=1e-5
tol.=1e-7
tol.=1e-2
tol.=1e-3
tol.=1e-4
tol.=1e-5
lmax=2 lmax=3lmax=1
tol.=1e-2
tol.=1e-3
tol.=1e-5
Minimum
Maximum
Remarks
Decreases with lmax.
Saturates with ǫtol
for a given lmax .

Interstitial
150 200 250 300 350
Number of interstitial basis functions
1e-08
1e-07
1e-06
1e-05
1e-04
0.001
Relativeerror
LAPWbasis
Minimum
Maximum
Remarks
Saturates with ǫtol for
a given lmax .
Decreases with
|Gmax|.

Interstitial
150 200 250 300 350
Number of interstitial basis functions
1e-08
1e-07
1e-06
1e-05
1e-04
0.001
Relativeerror
LAPWbasis
Minimum
Maximum
Remarks
Saturates with ǫtol for
a given lmax .
Decreases with |Gmax|.
Recipe
Choose max(ǫtol ) that
saturates
Choose |Gmax| so that
∆i ≈ ∆MT

Implementation Brillouin Zone Integration
The Polarization
Pij (q, ω) =
Z
BZ
X
nm

Mi
nm(k,q)[Mj
nm(k,q)]∗
ω−ǫmk−q+ǫnk+iη
−
Mi
nm(k,q)[Mj
nm(k,q)]∗
ω−ǫnk+ǫmk−q−iη
ﬀ
f (ǫk)[1 − f (ǫk−q)]d3
k

Brillouin Zone Integration
Pij (q, ω) =
Z
BZ
X
nm

Mi
nm(k,q)[Mj
nm(k,q)]∗
−
Mi
nm(k,q)[Mj
nm(k,q)]∗
ﬀ
f (ǫk)[1 − f (ǫk−q)]d3
k
q-dependent Linear Tetrahedron Method

Pij (q, ω) =
Z
BZ
X
nm

Mi
nm(k,q)[Mj
nm(k,q)]∗
−
Mi
nm(k,q)[Mj
nm(k,q)]∗
ﬀ
f (ǫk)[1 − f (ǫk−q)]d3
k
Linear Tetrahedron Method
P(q, ω) =
Z
BZ
X(k, q)
f (ǫk)[1 − f (ǫk−q)]
ω − ∆ǫ
d3
k
wT
ki ,q =
Z
ΩT
wi (k)
f (ǫk)[1 − f (ǫk−q)]
ω − ∆ǫ
d3
k wki ,q =
X
T∋ki
wT
ki ,q
P(q, ω) =
X
nm
X
i
X(ki , q)wnm(ki , q; ω)
J. Rath and A. J. Freeman, Phys. Rev. B 11, 2109 (1975)

Pij (q, ω) =
Z
BZ
X
nm

Mi
nm(k,q)[Mj
nm(k,q)]∗
−
Mi
nm(k,q)[Mj
nm(k,q)]∗
ﬀ
f (ǫk)[1 − f (ǫk−q)]d3
k

Pij (q, ω) =
Z
BZ
X
nm

Mi
nm(k,q)[Mj
nm(k,q)]∗
−
Mi
nm(k,q)[Mj
nm(k,q)]∗
ﬀ
f (ǫk)[1 − f (ǫk−q)]d3
k
q

Pij (q, ω) =
Z
BZ
X
nm

Mi
nm(k,q)[Mj
nm(k,q)]∗
−
Mi
nm(k,q)[Mj
nm(k,q)]∗
ﬀ
f (ǫk)[1 − f (ǫk−q)]d3
k

Brillouin Zone integration
Pij (q, ω) =
Z
BZ
X
nm

Mi
nm(k,q)[Mj
nm(k,q)]∗
−
Mi
nm(k,q)[Mj
nm(k,q)]∗
ﬀ
f (ǫk)[1 − f (ǫk−q)]d3
k
4 nodes 6 nodes 8 nodes 8 nodes

Test: Free electron gas
0 0.5 1 1.5 2 2.5 3 3.5
q/kF
0
0.2
0.4
0.6
0.8
1
Lindhardtfunction
exact
364 k-pts
540 k-pts
1368 k-pts

Test: Cu bandstructure
W L Γ X Z W K
-10
EF
10 LDA
GW

Test: Cu DOS
-8 -6 -4 -2 EF
2 4
Energy [eV]
0
1
2
3
4
5
6DOS
LDA
GW

Test: Al bandstructure
W L Γ X Z W K
-10
EF
10
LDA
GW

Implementation The Γ-point singularity
The Γ point singularity
The symmetrized dielectric matrix
Deﬁnition
˜εij (q, ω) =
lm
v
1
2
il (q)Plm(q, ω)v
1
2
mj (q)
The screened potential
Wij (q, ω) =
lm
v
1
2
il (q)˜ε−1
lm (q, ω)v
1
2
mj (q)

The screened Coulomb potential
v
1
2
ij (q → 0) =
v
s 1
2
ij
|q|
+ ˜v
1
2
ij (q)
Wij (q, ω) =
1
|q|2
W s2
ij (q, ω) +
1
|q|
W s1
ij (q, ω) + ˜Wij (q, ω)
Diverges!

The screened Coulomb potential
v
1
2
ij (q → 0) =
v
s 1
2
ij
|q|
+ ˜v
1
2
ij (q)
Wij (q, ω) =
1
|q|2
W s2
ij (q, ω) +
1
|q|
W s1
ij (q, ω) + ˜Wij (q, ω)
But can be integrated

Implementation Frequency convolution
Implementation
Frequency convolution
Σ(r1, r2; ω) =
i
2π
Z
G0(r1, r2; ω + ω′
)W0(r2, r1; ω‘)dω′
⇓ Analytic continuation
Σ(r1, r2; iω) =
i
2π
Z
G0(r1, r2; iω + iω′
)W0(r2, r1; iω‘)diω′

Implementation
Σ(r1, r2; ω) =
i
2π
Z
G0(r1, r2; ω + ω′
)W0(r2, r1; ω‘)dω′
Σ(r1, r2; iω) =
i
2π
Z
G0(r1, r2; iω + iω′
Σnk(iω) = −
X
q
X
ij
X
n′
Mi
nn′ (k, q)
1
π
∞Z
0
Wij (q; iω′)dω′
(iω − ǫn,k′ )2 + ω′2
M∗j
n′n
(k, q)

Implementation
Σ(r1, r2; ω) =
i
2π
Z
G0(r1, r2; ω + ω′
)W0(r2, r1; ω‘)dω′
Σ(r1, r2; iω) =
i
2π
Z
G0(r1, r2; iω + iω′
Σnk(iω) = −
X
q
X
ij
X
n′
Mi
nn′ (k, q)
1
π
∞Z
0
Wij (q; iω′)dω′
(iω − ǫn,k′ )2 + ω′2
M∗j
n′n
(k, q)
Pad´e Approximant
Σnk(iω) =
PN
j=0 aj (iω)j
PN+1
j=0 bj (iω)j
Σnk(ω) =
PN
j=0 aj ωj
PN+1
j=0 bj ωj

Implementation Summary
G0W0@Wien2k: A FP-(L)APW+lo + G0W0 code
Flowchart
Wien2k Begin
ψkn χiq
ǫDFT
kn Mi
nm(k, q) vij (q)
Pij (q, ω)
εij (q, ω)
Wij (q, ω)
V xc
k,n Σnn(k, ω)
ǫ
qp
k,n
End
Code keywords
Based on FP-(L)APW+lo
Mixed Basis
Linear Tetrahedron
Method
Reciprocal space
Imaginary frequencies
Excelent results

Implementation Convergence Tests
Silicon: Convergence tests
Number of frequencies
10 15 20 25
Nr. of frequencies
0.878
0.879
0.88
0.881
0.882
0.883
0.884
Eg
[eV]
Used paremeters
16 frequencies

Number of excited states
8 27 64 125
Nr. of k-points
0.45
0.46
0.47
0.48
0.49
0.5
0.51
0.52
0.53
∆Eg
[eV]
Used paremeters
16 frequencies
64 k-points

Number of k-points
0 50 100 150 200 250
Number of unoccupied bands
0.95
1
1.05
1.1
BandGap
Used paremeters
16 frequencies
64 k-points
∼ 200 unocc. bands

Results Bandgaps
Results I:
Bandgaps Si
GaAs
zGaN
ZnS
C
MgO
NaCl
CaO
0 2 4 6 8
Experimental Eg
[eV]
0
2
4
6
8
10
CalculatedEg
[eV]
G0
W0
All electron
G0
W0
Pseudopotentials
G0
W0
@Wien2k
LDA
- F. Aryasetiawan and O.
Gunnarson, Rep. Prog. Phys.
61, 237 (1998).(and Refs.)
- T. Kotani and M. van
Schilfgaarde, Solid State Comm.
121, 461 (2002).
- C. Friedrich et al, Phys. Rev. B
74, 045104 (2006).

Results Bandstructures
Results II:
Band diagrams: Silicon
W ΓL Λ ∆ X Z W K
-10
-5
εF
5
10
1
LDA
GW

Core-valence interaction Motivation
Motivation
G0W0 Si bandgap
1990 2000
Publication year
0
0.5
1
1.5
Eg
[eV]
Pseudopotentials
All electron

Core-valence interaction Motivation
Debate
G0W0 Si bandgap
1990 2000 2010
Publication year
0
0.5
1
1.5
Eg
[eV]
Pseudopotentials
All electron
?
- M. Tiago et al, Phys. Rev. B 69, 125212 (2004).
- C. Friedrich et al, Phys. Rev. B 74, 045104 (2006).
Pseudopotentials
systematically larger
gaps
in better agreement
with expeeriment
All electron
Benchmark for ab-initio
calculations
- A. Fleszar, Phys. Rev. B 64, 245204
(2001).
- T. Kotani and M. van Schilfgaarde,
Solid State Comm. 121, 461 (2002).
- W. Ku and A. Eguiluz, Phys. Rev. Lett.
89,126401 (2002).

Core-valence interaction Core-valence Partitioning
Core-valence partitioning
Kohn-Sham equation
HKS Ψnk = ǫnkΨnk
All electron
HKS = T + Vnuc + Vh + Vxc[n]
Pseudopotentials
HKS = T + Vps + Vh + Vxc[ñv + ñc]
Vxc[n] = Vxc[ñv + ñc] + Vxc[ncore − ñc] ⊂ Veff
G0W0 correction
ǫqp
nk = ǫKS
nk + ∆ǫnk
All electron
∆ǫnk = ℜ (Σnk [{Ψnk; Ψc}]) − V xc
nk [n]
Pseudopotentials
∆ǫnk = ℜ(Σnk[{˜Ψnk}])−V xc
nk [ñv +ñc]
ℜ (Σnk [{Ψnk; Ψc}]) = ℜ(Σnk[{˜Ψnk}]) + V xc
nk [ncore − ñc] ⊂ Veff

Core-valence interaction Different Approaches
Different Approaches
G0W0 correction
All electron:
ǫqp
nk = ǫKS
nk + ℜ (Σnk [{Ψnk; Ψcore}]) − V xc
nk [n]
Pseudopotentials:
ǫqp
nk = ǫKS
nk + ℜ(Σnk[{˜Ψnk}]) − V xc
nk [ñval ]
Valence only:
ǫqp
nk = ǫKS
nk + ℜ (Σnk [{Ψnk}]) − V xc
nk [nval ]
Separated terms
Σnk = Σx
nk + Σc
nk

Core-valence interaction Silicon
Results
Si: Matrix elements
Highest occ. state at Γ Lowest unocc. state at X

Results
Si: Matrix elements
Remarks
ΣC : PP ≈ Val ≈ AE
ΣX : PP ≈ Val = AE
VXC : PP ≈ Val = AE
Core-valence lin.
ΣC : Small effect
ΣX : Large effect
VXC : Large effect

Results
Si: Matrix elements
Remarks
ΣX − VXC : PP ≈ Val ≈
AE
Core-valence lin.
ΣC : Small effect
ΣX : Large effect
VXC : Large effect
ΣX − VXC :Small effect

Results
Si gap: G0W0 correction
Remarks
∆Eg : PP > Val > AE
Core-valence lin.
∆Eg : Small eﬀect.

Core-valence interaction GaAs
Results
GaAs: Matrix elements
Highest occ. state at Γ Lowest unocc. state at Γ

Results
Remarks
Core-valence lin.
ΣC : Small effect
ΣX : Large effect
VXC : Large effect

Results
Remarks
ΣX − VXC : PP ≈ Val =
AE
Core-valence lin.
ΣC : Small effect
ΣX : Large effect
VXC : Large effect
ΣX − VXC :Large effect

Results
GaAs gap: G0W0 correction
Remarks
∆Eg : PP < Val < AE
Core-valence lin.
∆Eg : Large eﬀect.
Reduces the correction!!

Conclusions
Core-valence partitioning:
Strong changes in ΣX and VXC
Small changes in ΣC
∆Eg : Small changes in Si. Large in GaAs
Does NOT systematically increase the G0W0-correction.
“Pseudoization” also plays an important role.

Core-valence interaction Semicore States
Semicore states
Semicore states binding energies
MgO 6p-5p1s transition energy
TotalW L Λ Γ ∆ X ZW K
-50
-40
-30
-20
-10
EF
10
Energy[eV]
sMg
pMg
s0
pO
LDA:=43.55eV
GW:=52.41eV
Excitation energy [eV]
ROHF(∆SCF)1
54.6
CASTP21
53.8
LDA2
43.5
G0W 2
0 52.4
Experiment3
53.4
1.- C. Sousa et al. Phys. Rev. B 62,
10013 (2000).
2.- This work.
3.- W. L. OBrien et al. Phys. Rev. B
44, 1013 (1991).

f-electron systems
f-electron systems
Motivation
Two strongly interrelated subsystems
itinerant spd states
strongly localized f states
Intriguing physics
Heavy fermion
Kondo eﬀect
Etc...
Challenge to ﬁrst-principle calculations:
LDA/GGA good for itinerant electrons
Fails for f -electrons

f-electron systems
(no)f-electron systems
CeO2
-6 -4 -2 0 2 4 6 8 10 12
Energy [eV]
0
5
10
15
DOS LDA
LDA-G0
W0
XPS+BIS
XPS+XAS
E. Wuilloud et al. Phys. Rev. Lett 53, 202 (1984)
D. R. Mullins et al. Surf. Sci. 409, 307 (1998)

f-electron systems
(no)f-electron systems
Bandgaps
ZrO2
HfO2
CeO2
p-f
CeO2
p-d
ThO2
0
1
2
3
4
5
6
7
Eg
[eV]
LDA
G0
W0
Expt.

f-electron systems
f-electron systems
Ce2O3
-10 -8 -6 -4 -2 0 2 4 6 8 10
Energy [eV]
0
5
10
DOS
Expt.
LDA
XPS XAS

f-electron systems
f-electron systems
Ce2O3
-10 -8 -6 -4 -2 0 2 4 6 8 10
Energy [eV]
0
5
10
DOS
Expt.
LDA
LDA+U
XPS XAS

f-electron systems
f-electron systems
Ce2O3
-10 -8 -6 -4 -2 0 2 4 6 8 10
Energy [eV]
0
5
10
DOS
Expt.
LDA
LDA+U
LDA+U+G0
W0
XPS XAS

f-electron systems
f-electron systems
Bandgaps
La2
O3
Ce2
O3
Pr2
O3
Nd2
O3
0
1
2
3
4
5
6Eg
[eV]
Expt.
LDA+U
G0
W0

Conclusions
Ongoing work: Spin polarized metals
Test: Ni
W L Γ X ZW K
-10
-5
EF
5
Energy[eV]
W L Γ X ZW K
UP DOWN

Conclusions
Conclusions
GW@Wien2k
Reliable results.
Wide range of materials.
sp semiconductors
f -electron systems with empty or full f -shells
metals
spin polarized materials
LDA+U+G0W0

Conclusions
Conclusions
Most important

Conclusions
Conclusions
Most important
Now we are enjoying it!!!

Conclusions
Last but not least
Future plans
Anisotropy of εmacro
Half metals
Eﬃciency Improvement
COHSEX@Wien2k + GW
BSE
QPscGW

Conclusions
Acknowledgements
Xinzheng Li (FHI, Berlin): Code development, LTM library.
Dr. Hong Jiang (FHI, Berlin): Code improvement, Spin polarization,
LDA+U.
Prof. Claudia Ambrosch-Draxl (MUL; Austria): Wien2k interface and
more...
Christian Meisenbichler (MUL; Austria): MPI Paralelization
Patrick Rinke and Christoph Freysoldt (FHI, Berlin):
Pseudopotentials, etc..
The boss
Matthias Scheﬄer

The all-electron GW method based on WIEN2k: Implementation and applications.

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